On Matveev’s question about virtual 3-manifolds
E. A. Fominykh, A. V. Malyutin, and A. Yu. Vesnin
Abstract. S. V. Matveev introduced the concept of virtual 3-manifolds and
posed a problem, whether the natural map from the set of nondegenerate virtual 3-manifolds to the set of 3-manifolds with RP 2 -singularities is injective.
We show that this map is not injective and present an infinite family of virtual
3-manifolds corresponding to the same 3-manifold with RP 2 -singularities.
Matveev’s virtual 3-manifolds can be naturally defined either in terms of
spines or in the ‘dual’ language of singular triangulations. Contrary to the original
definition via spines, we start with triangulations.
Face identification schemes. Let n be a positive integer, let D = {∆1 , . . . , ∆n } be
a set of n disjoint tetrahedra, and let Φ = (φ1 , . . . , φ2n ) be a collection of 2n affine
homeomorphisms between the facets (i. e., triangular faces) of tetrahedra in D
such that each facet has a unique counterpart. Following [Mat03, p. 11], we refer
to the pair (D, Φ) as to a face identification scheme (a scheme). The quotient space
Q := Q(D, Φ) of the scheme (D, Φ) is defined as the space obtained from D by
identification of faces via the homeomorphisms in Φ. For any scheme, Q is either
a genuine or a singular 3-manifold (see [Sei33] or [Mat03, Proposition 1.1.23]). All
singular points of Q correspond either to vertices or to barycenters of edges of
the tetrahedra in D. The latter happens if the quotient map folds an edge so that
symmetric points (with respect to the barycenter of the edge) have the same image.
If a singularity point x in Q corresponds to the barycenter of an edge, then the link
of x is himeomorphic to the projective plane RP 2 , i. e., x is an RP 2 -singularity.
Pseudo-Pachner move. In Euclidean three-space, let P be the convex hull of five
points that are in general position. Assume that P is not a tetrahedron. Then P
is the union of two geometric tetrahedra (∆1 and ∆2 , say) and at the same time
P is the union of three geometric tetrahedra with disjoint interiors (∆3 , ∆4 , and
∆5 , say). Let
τ = ∂∆1 ∪ ∂∆2
Supported by RFBR grant 16-01-00609.
and
τ 0 = ∂∆3 ∪ ∂∆4 ∪ ∂∆5 .
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E. A. Fominykh, A. V. Malyutin, and A. Yu. Vesnin
If Q := Q(D, Φ) is the quotient space of the scheme (D, Φ), we denote by
σ(Q) the image in Q of boundaries of tetrahedra in D (2-skeleton of (D, Φ)).
We say that two schemes (D, Φ) and (D0 , Φ0 ) are related by a pseudo-Pachner
move if there exists a homeomorphism h : Q → Q0 between the quotient spaces
Q := Q(D, Φ) and Q0 := Q(D0 , Φ0 ) and a map f : P → Q0 such that
– the restriction of f to the interior of P is an embedding;
– h(σ(Q)) ∩ f (P ) = f (τ ) and σ(Q0 ) ∩ f (P ) = f (τ 0 );
– h(σ(Q)) \ f (P ) = σ(Q0 ) \ f (P ), that is, h(σ(Q)) and σ(Q0 ) do coincide
outside f (P ).
Definition. Virtual 3-manifolds. We say that two face identification schemes are
equivalent if they are related by a chain of pseudo-Pachner moves. A virtual 3manifold is an equivalence class of schemes. We say that a virtual 3-manifold
is degenerate if it corresponds to a scheme with precisely one tetrahedron.
Truncated quotient spaces as underlying spaces of virtual 3-manifolds. The truncated quotient space Qt := Qt (D, Φ) of the scheme (D, Φ) is defined as the space
obtained from the quotient space Q := Q(D, Φ) by deleting small open proper
neighborhoods of points in Q that are images of vertices of the tetrahedra in D.
Truncated quotient spaces of equivalent schemes are obviously homeomorphic. By
the underlying space Qt (V ) of a virtual 3-manifold V we will mean the truncated
quotient space of schemes in V . Thus, the underlying space of a virtual 3-manifold
is a 3-manifold with RP 2 -singularities.1 The following facts follow from key results
of the special spine theory (see [Mat03, Mat09]).
Corollary 1. (1) Each compact connected 3-manifold M with RP 2 -singularities
and nonempty boundary is the underlying space of a virtual 3-manifold.
(2) If M is, moreover, genuine, then it is the underlying space of a unique nondegenerate virtual 3-manifold.
Problem. The following natural question arises (see [Mat09]). Do the second statements of Corollary 1 holds for 3-manifolds with RP 2 -singularities? That is, are
there distinct nondegenerate virtual 3-manifolds with the same connected underlying space?
We present the following theorem that answers the question above.
Theorem 1. The cone C(RP 2 ) = (RP 2 × [0, 1])/(RP 2 × {0}) over the projective
plane RP 2 is the underlying space for an infinite number of pairwise distinct virtual
3-manifolds.
In order to explain the idea of the proof of Theorem 1, we introduce the
concepts of 3-manifolds with traced RP 2 -singularities.
1A
compact three-dimensional polyhedron W is called a 3-manifold with RP 2 -singularities if
the link of any point of W is either a 2-sphere, or a 2-disc, or RP 2 .
On Matveev’s question about virtual 3-manifolds
3
3-manifolds with traced RP 2 -singularities. A 3-manifold with traced RP 2 -singularities is a pair (W, I), where W is a 3-manifold with RP 2 -singularities and I is
a subpolyhedron of W with the following properties:
– I is a disjoint union of arcs (these arcs are called traces),
– the number of components in I equals the number of singular points in W ,
– each component of I emanates from the boundary of W and ends at a
singularity point of W .
We describe two interpretations for 3-manifolds with traced RP 2 -singularities.
3-manifolds with Möbius singularities. We say that a compact 3-dimensional polyhedron W is a 3-manifold with Möbius singularities if the link of any point of W
is either a 2-sphere, or a 2-disc, or a Möbius band. (The set of points of W whose
links are 2-discs or Möbius bands form the boundary ∂W of W , so that the singularities of W are contained in its boundary.) Collapsing traces to points, we see
that the set of 3-manifold with traced RP 2 -singularities is in a natural one-to-one
correspondence with the set of 3-manifolds with Möbius singularities.
3-manifolds decorated with orientation-reversing curves. A closed simple curve γ
on a surface is orientation-reversing if a tubular neighbourhood of γ is a Möbius
band. If W is a genuine 3-manifold and C is a collection of closed simple pairwise disjoint orientation-reversing curves on the boundary ∂W , then collapsing
each curve in C to a point transforms W to a 3-manifolds with Möbius singularities (each curve of C gives a singularity point). Thus, the set of 3-manifolds
equipped with orientation-reversing curves is another interpretation for the set of
3-manifolds with traced RP 2 -singularities.
From virtual 3-manifolds to 3-manifolds with traced RP 2 -singularities. Each virtual 3-manifold is naturally assigned with a 3-manifold with traced RP 2 -singularities. Indeed, observe that all singular points of the truncated quotient space
Qt := Qt (D, Φ) of the scheme (D, Φ) are RP 2 -singular points corresponding to
barycenters of edges of the tetrahedra of D. The quotient map folds the edges
that give singular points so that symmetric points (with respect to the barycenters) of each such folding edge have the same image. We observe that the pair
(Qt , I), where I is the image in Qt of all folding edges, is a 3-manifold with
traced RP 2 -singularities. The described correspondence (D, Φ) 7→ (Qt , I) associates a 3-manifold with traced RP 2 -singularities to each scheme. Now, observe
that any pseudo-Pachner move changes neither Qt nor the subset in Qt formed by
the images of folding edges, so that equivalent schemes yield the same 3-manifold
with traced RP 2 -singularities. This gives a map (which we denote by F ) from the
set of virtual 3-manifolds to the set of 3-manifolds with traced RP 2 -singularities.
Note that by construction, forgetting the traces in F (V ), where V is a virtual
3-manifold, converts F (V ) to the underlying space of V .
Theorem 2. If (M, I) is a compact connected 3-manifold with traced RP 2 -singularities and nonempty boundary, then there exists a virtual 3-manifold V such that
F (V ) = (M, I).
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E. A. Fominykh, A. V. Malyutin, and A. Yu. Vesnin
Lemma 1. The cone C(RP 2 ) = (RP 2 ×[0, 1])/(RP 2 ×{0}) over the projective plane
RP 2 is the base space for an infinite number of pairwise distinct 3-manifolds with
traced RP 2 -singularities. In other words, the singularity point in C(RP 2 ) can be
traced in an infinite number of pairwise nonequivalent ways.
Ideas of proofs. Theorem 1 follows from Theorem 2 and Lemma 1.
Theorem 2 can be proved in terms of special spine theory on the base of
results established in [Mat03]. Given a compact connected 3-manifold with traced
RP 2 -singularities (M, I), we delete from M open proper neighbourhoods of I.
The obtained genuine 3-manifold MI ⊂ M possesses a special spine S (see [Cas65]
or [Mat03, Theorem 1.1.13]). Observe that ∂MI \ ∂M is the union of disjoint open
Möbius bands. Let C ⊂ ∂MI be the union of center curves of these Möbius bands.
A natural retraction p : MI → S sends C to a collection of curves on S. We attach
to S new unthickenable 2-cells along p(C) and show that the obtained special
polyhedron represents a virtual 3-manifold V with F (V ) = (M, I).
In order to prove Lemma 1, consider ‘knotted’ traces and use the branched
double covering of C(RP 2 ) by the 3-ball (ramified over the singularity point). References
[Cas65] B. G. Casler, An imbedding theorem for connected 3-manifolds with boundary,
Proc. Amer. Math. Soc., 16, 559–566 (1965).
[VTF16] A. Yu. Vesnin, V. G. Turaev, E. A. Fominykh, Complexity of virtual 3-manifolds,
Mat. Sb., 207:11 (2016), 4–24; Sb. Math., 207:11 (2016), 1493–1511.
[Mat03] S. V. Matveev, Algorithmic topology and classification of 3-manifolds, Algorithms and Computation in Mathematics, vol. 9, Springer-Verlag, Berlin, 2003.
[Mat09] S. V. Matveev, Virtual 3-manifolds, Siberian electronic mathematical reports 6
(2009), 518–521.
[Sei33] H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math., 60
(1933), 147–238.
E. A. Fominykh
Chelyabinsk State University and Krasovskii Institute of Mathematics and Mechanics
Chelyabinsk, Russia
e-mail: [email protected]
A. V. Malyutin
St. Petersburg Department of Steklov Institute of Mathematics
St. Petersburg, Russia
e-mail: [email protected]
A. Yu. Vesnin
Sobolev Institute of Mathematics
Novosibirsk, Russia
e-mail: [email protected]